Do Now In BIG CLEAR numbers, please write your height in inches on the index card.

Chapter 3: Modeling Distributions of Data Section 3.1 Describing Location in a Distribution

Chapter 3 Modeling Distributions of Data 3.1 Describing Location in a Distribution 3.2 Normal Distributions

Describing Location in a Distribution Measuring Position: Percentiles One way to describe the location of a value in a distribution is to tell what percent of observations are less than it. Describing Location in a Distribution Definition: The pth percentile of a distribution is the value with p percent of the observations less than it. “You’re the 80th percentile for the class height!”

Example Jenny earned a score of 86 on her test. How did she perform relative to the rest of the class? Her score was greater than 21 of the 25 observations. Since 21 of the 25, or 84%, of the scores are below hers, Jenny is at the 84th percentile in the class’s test score distribution. 6 7 7 2334 7 5777899 8 00123334 8 569 9 03

Try This! Lynette, a student in the class, is 65 inches tall. Is she tall or short relative to her classmates? What is your percentile of the distribution of heights in this class? What height in the dot plot above is at the 80th percentile? We can see that her height is below average, but how much below average is it?

Describing Location in a Distribution Measuring Position: z-Scores A z-score tells us how many standard deviations from the mean an observation falls, and in what direction. Describing Location in a Distribution Definition: If x is an observation from a distribution that has known mean and standard deviation, the standardized value of x is: A standardized value is often called a z-score.

What About Jenny? Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is 6.07. What is her standardized score?

Describing Location in a Distribution Using z-scores for Comparison Describing Location in a Distribution We can use z-scores to compare the position of individuals in different distributions. Jenny earned a score of 86 on her statistics test. The class mean was 80 and the standard deviation was 6.07. She earned a score of 82 on her chemistry test. The chemistry scores had a fairly symmetric distribution with a mean 76 and standard deviation of 4. On which test did Jenny perform better relative to the rest of her class?

Where does Lynette's height fall relative to the mean? Lynette=65 inches

Standardized values and z-scores We can also use the z-score to compare the locations of individuals in different distributions or different scales. Example : SAT vs. ACT Sophia scores 660 on the SAT Math test. The distribution of SAT scores has a mean of 500 and standard deviation of 100. Jim takes the ACT Math test and scores 26. ACT scores have a 
mean of 18 and standard deviation of 6. Assuming that both tests measure the same kind of ability, who did better? Find the z-score from each to compare...

Standardized values and z-scores Example: Finding the value What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 5?

Describing Location in a Distribution Density Curves In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy. Describing Location in a Distribution Exploring Quantitative Data Always plot your data: make a graph. Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers. Calculate a numerical summary to briefly describe center and spread. 4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

Describing Location in a Distribution Density Curve Definition: A density curve is a curve that is always on or above the horizontal axis, and has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval. Describing Location in a Distribution The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.

Figure 3.4 (a) The proportion of scores less than 
6.0 from the histogram is 0.303. (b) The proportion of scores less than 
6.0 from the density curve is 0.293.

MEDIAN AND QUARTILES The median of a density curve is the equal-areas point, the point with half the area under the curve to its left and the remaining half of the area to its right. The quartiles divide the area under the curve into quarters. ¼ of the area under the curve is to the left of the first quartile, and ¾ of the area is to the left of third quartile.

MEAN The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median and mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

If the density curve is perfectly symmetric, the mean = the median.

The median divides the curves area in half The median divides the curves area in half. Mean is the balance point; where the curve would balance if its made of solid material. The mean is pulled towards the skew or long tail.

(sigma) Standard Deviation New Symbols Used to discuss a sample (mu) Mean Used to discuss the whole population. We use these symbols when we are discussing density curves and more. (sigma) Standard Deviation

Example 1: A uniform distribution Why is the total area under this curve equal to 1? What percent of the observations lie above 0.8? What percent of the observations lie below 0.6? What percent of the observations lie between 0.25 and 0.75? What is the mean, μ of this distribution?

Example 2 An unusual “broken line” density curve. Use areas under this density curve to find the proportion of observations within the given interval.

Example 4 Consider a uniform distribution that outputs numbers between 0 and 2. Find P (x < 0.5) Find P (x > 0.7) Find P (0.2 < x < 1.2)

Describing Location in a Distribution Describing Density Curves Our measures of center and spread apply to density curves as well as to actual sets of observations. Describing Location in a Distribution Distinguishing the Median and Mean of a Density Curve The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

Section 3.1 Describing Location in a Distribution Summary In this section, we learned that… There are two ways of describing an individual’s location within a distribution – the percentile and z-score. A cumulative relative frequency graph allows us to examine location within a distribution. It is common to transform data, especially when changing units of measurement. Transforming data can affect the shape, center, and spread of a distribution. We can sometimes describe the overall pattern of a distribution by a density curve (an idealized description of a distribution that smooths out the irregularities in the actual data).